Object relationships appear intact for some child objects, non-existent for others
Ok, I figured this out and I feel (kind of) dumb now.  

Some objects were positioned in edit mode, while others were positioned in object mode.  

The ones that were positioned in edit mode still have their object origin at (0,0,0) with vertices positioned away from the origin. 

The ones that were 'misbehaving' were positioned in object mode, and had their origin at some non-zero position. 

In both cases, the relationship line was drawn from the origin to the object position, but in the first case, it wasn't visible (since the object position and origin were the same point).